Q.If the straight lines $\dfrac{x-1}{2}=\dfrac{y+1}{\lambda}=\dfrac{z}{2}$ and $\dfrac{x+1}{5}=\dfrac{y+1}{2}=\dfrac{z}{\lambda}$ are coplanar, find $\lambda$ and equations of the planes containing these two lines.

Step 1. Data. $\vec a=(1,-1,0),\ \vec b=(2,\lambda,2);\quad \vec c=(-1,-1,0),\ \vec d=(5,2,\lambda)$.

Step 2. Coplanarity determinant. $\vec c-\vec a=(-2,0,0)$.

Step 3. Set to zero. $-2\lambda^2+8=0\Rightarrow\lambda^2=4\Rightarrow\lambda=\pm2$.

Step 4. Case $\lambda=2$: normal $=\vec b\times\vec d$ with $\vec b=(2,2,2),\vec d=(5,2,2)$ (use simplified $\vec b\propto(1,1,1)$):

Plane through $\vec a=(1,-1,0)$, normal $(0,1,-1)$: $0(x-1)+1(y+1)-1(z-0)=0\Rightarrow y-z+1=0$. …

Coplanarity determinant with one simple row, solve for $\lambda$, t …

• Solving only for $\lambda$ and forgetting the question also asks for both planes …